I can generalize the notion of a space over $X$ (where $X$ is a based and connected space) to the notion of a spectrum over $X$ by considering functors of quasicategories $X\to Mod_\mathbb{S}$. In the language of recent work of Ando, Blumberg and Gepner such functors can be thought of as bundles (i.e. locally constant sheaves) of $\mathbb{S}$-modules over $X$, or as $X$-parameterized spectra. In the case that we have a *space* parameterized by $X$, we know relatively obviously that $\Omega X$ acts on the fiber over the base point. This of course comes from just thinking about the fact that we can let the fiber travel along the path traced out by an loop in $X$.

More generally however, it seems that it should be true that $\Omega X$ acts on the "fiber" of parameterized spectrum over $X$. How can I make a formal argument to this effect, which perhaps relies only on category-theoretic considerations? In particular, if I'm not mistaken, there is an equivalence between "spectra over $X$" and "spectra with an $\Omega X$-action" for which "taking the fiber" is one direction.

In general I feel like this should be very generally true for the category of functors $X\to \mathcal{C}$ for any (presentable?) quasicategory $\mathcal{C}$.

connectedspace. $\endgroup$ – Qiaochu Yuan Jun 14 '15 at 21:21